Learning filtered discretization operators: non-intrusive versus intrusive approaches

TL;DR

This paper introduces a novel filtering approach for discretized equations, comparing intrusive and non-intrusive methods, and evaluates their accuracy and stability in modeling multi-scale phenomena like turbulent flows.

Contribution

It proposes a new discretization-first filtering method and compares three operator inference techniques, highlighting their advantages and limitations.

Findings

Explicit reconstruction and derivative fitting yield small errors

Trajectory fitting requires extensive training to match performance

Explicit reconstruction is more prone to instabilities

Abstract

Simulating multi-scale phenomena such as turbulent fluid flows is typically computationally very expensive. Filtering the smaller scales allows for using coarse discretizations, however, this requires closure models to account for the effects of the unresolved on the resolved scales. The common approach is to filter the continuous equations, but this gives rise to several commutator errors due to nonlinear terms, non-uniform filters, or boundary conditions. We propose a new approach to filtering, where the equations are discretized first and then filtered. For a non-uniform filter applied to the linear convection equation, we show that the discretely filtered convection operator can be inferred using three methods: intrusive (`explicit reconstruction') or non-intrusive operator inference, either via `derivative fitting' or `trajectory fitting' (embedded learning). We show that explicit reconstruction and derivative fitting identify a similar operator and produce small errors, but that trajectory fitting requires significant effort to train to achieve similar performance. However, the explicit reconstruction approach is more prone to instabilities.